If $x = a$ makes the denominator $q(x)$ zero and the numerator $p(x)$ non-zero, then there's a potential vertical asymptote at $x = a$.

It might indicate a 'hole' in the graph rather than a vertical asymptote.

If the multiplicity of a zero in the denominator is greater than in the numerator, there's a vertical asymptote. Higher difference means faster approach to infinity.

As $x$ approaches a vertical asymptote from the left or right, the function approaches either positive or negative infinity.

One-sided limits approaching a vertical asymptote will tend towards $\infty$ or $-\infty$.

Vertical Asymptotes: Occur when the denominator is zero, and the numerator is non-zero. Function approaches infinity. | Holes: Occur when both numerator and denominator are zero. Represent removable discontinuities.

1. Factor the numerator and denominator. 2. Identify zeros of the denominator. 3. Check if those zeros are also zeros of the numerator. 4. If a zero is only in the denominator, there's a vertical asymptote there.

1. Find the vertical asymptote. 2. Evaluate the one-sided limits as $x$ approaches the asymptote from the left and right. 3. Determine if the function approaches $\infty$ or $-\infty$ from each side.

1. Factor the numerator and denominator. 2. If a factor cancels out from both, there is a hole at the x-value that makes that factor zero.

1. Factor: $r(x) = \frac{(x-1)(x+1)}{(x-1)(x-1)}$. 2. Simplify: $r(x) = \frac{x+1}{x-1}$. 3. Vertical asymptote: $x=1$.